$\text{ If }f\left(x\right)=|x-2|,g\left(x\right)=f\left(f\right(x\left)\right)\text{, then for }x>4,g^1\left(x\right)$

$\begin{array}{l}\text{ if }x>4\text{ then }|x-2|=x-2\\ g\left(x\right)=f(x-2)\\ =|x-2-2|\\ =|x-4|\\ \therefore g\left(x\right)=x-4\\ \Rightarrow g^1\left(x\right)=1\end{array}$